| L(s) = 1 | + 1.00·3-s + 6.07·5-s − 26.7·7-s − 25.9·9-s + 82.1·13-s + 6.09·15-s − 103.·17-s − 28.7·19-s − 26.7·21-s + 104.·23-s − 88.1·25-s − 53.1·27-s + 13.3·29-s + 333.·31-s − 162.·35-s + 71.0·37-s + 82.4·39-s − 319.·41-s + 103.·43-s − 157.·45-s + 402.·47-s + 370.·49-s − 104.·51-s + 14.0·53-s − 28.8·57-s + 246.·59-s + 832.·61-s + ⋯ |
| L(s) = 1 | + 0.193·3-s + 0.543·5-s − 1.44·7-s − 0.962·9-s + 1.75·13-s + 0.104·15-s − 1.48·17-s − 0.347·19-s − 0.278·21-s + 0.945·23-s − 0.704·25-s − 0.378·27-s + 0.0855·29-s + 1.93·31-s − 0.783·35-s + 0.315·37-s + 0.338·39-s − 1.21·41-s + 0.365·43-s − 0.523·45-s + 1.25·47-s + 1.07·49-s − 0.286·51-s + 0.0364·53-s − 0.0670·57-s + 0.543·59-s + 1.74·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.734163544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734163544\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.00T + 27T^{2} \) |
| 5 | \( 1 - 6.07T + 125T^{2} \) |
| 7 | \( 1 + 26.7T + 343T^{2} \) |
| 13 | \( 1 - 82.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 13.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 333.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 71.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 103.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 402.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 14.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 832.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 63.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 408.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 855.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 524.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 392.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.505695385372342923618363623633, −8.834269664327662449377740556657, −8.267315154384620862726976806126, −6.68692521794617529459182580924, −6.36792157134079965786272372987, −5.52954255328088213465069269188, −4.11390708214282107388714682594, −3.18518243345914781685943733120, −2.29474705614730014300120085406, −0.67959518183271146473886493740,
0.67959518183271146473886493740, 2.29474705614730014300120085406, 3.18518243345914781685943733120, 4.11390708214282107388714682594, 5.52954255328088213465069269188, 6.36792157134079965786272372987, 6.68692521794617529459182580924, 8.267315154384620862726976806126, 8.834269664327662449377740556657, 9.505695385372342923618363623633
